Nonlinear Finite Element Analysis of Concrete Structures

— 1 f> -> _
A transformation to polar coordinates r and 6, fig. 1, is performed
by the substitutions x = rsinø and y = rcosB, and using
3
the identify cos39 = 4cos 6-3 cosø we derive
w = -^ (JJ h 3 - hr 2 - r 3 cos39j (A-l)
The contour lines of the deflected membrane in the polar coordinates
r and 0 are determined by this equation treating w as
a constant. It is obvious that these contour lines are smooth
and convex and varying between the equilateral triangle and
a circle. To determine these contour lines we note that the
2
maximum deflection w = ph /27S occurs at r = 0 and disregardmax
L
^
ing in the following the point r = 0, the positive constant D
is defined by
D =
3(# - fO
Introducing this constant in eq. (1) and rearranging this equation
we obtain
1_ 3_ 1_ 3cos38 _
3 ~ 2 2
i- J D r hD^
Solving this cubic equation by standard methods it appears that
the roots of interest are only
A = ^ = K 1 cos ^ Arccos(K 2 cos39) j ;
cos36 _> O
A = - = K 1 cos ^ - ^ Arccos(-K 2 cos38) ; cos36